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Lecture 15

# Lecture 15 - Identical particles Bosons and fermions...

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Identical particles Bosons and fermions

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Two-particle systems 2 2 2 2 1 2 1 1 2 2 int 1 2 1 2 1 1 ( , ) ( , ) ( , , ) 2 2 H V V V m m σ σ σ σ = - ∇ - - ∇ + + + r r r r h h The system consisting of two particles is described by a wavefunction dependent on coordinates and spins of both particles: 1 2 1 2 ( , , ) t σ σ Ψ r r The Hamiltonian will include kinetic and potential energies of each particle, as well as possible potential energy of interaction between them. In the absence of interaction between particles ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 1 1 2 2 2 2 2 1 , , , , , ( , ) 2 i i i i i n i ni n i V E m σ σ ψ σ σ ψ σ ψ σ σ ψ ψ = - ∇ + = r r r r r r r h int 1 2 1 2 ( , , , ) 0 V t σ σ = r r Coordinates of the particles can be separated, i.e. the wave function is presented as a product of wave functions of individual particles satisfying single-particle SE Spin dependence of the single particle energies means that those energies are two- by-two matrices with entries different for different states of the spin.
Distinguishable particles Consider a system of two particles, which can occupy one of two single-particle states: 1 2 / / 1 , 2 iE t iE t e e - - h h There are multiple two-particle states corresponding to total energy 1 2 E E + ( 29 ( 29 ( 29 ( 29 1 2 1 2 1 2 / / 1 2 2 1 / 1 1 1 2 2 1 1 2 ; 1 2 1 2 1 2 i E E t i E E t i E E t e e a b e - + - + - + + h h h The last line here represent an arbitrary combination of degenerate states listed in the first line. If such a state is realized, we cannot say, which particle is in which state, there are certain probabilities to find either of them in either of the states. Such states are called entangled and are not easy to create.

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Identical particles In classical mechanics, even if you are dealing with identical particles, you can still distinguish between them, following, for instance, each particle’s path. In quantum mechanics identical particles are indistinguishable. Indeed, imaging you conduct a two- slit experiment with two particles. Since you don’t know through which slit an individual particle passes, you can not distinguish them in principle. Even if you try to track positions of each particle, you make its momentum so undetermined that you again have no idea, if a particle found at the time of the next measurement is the same or a different. Thus, quantum mechanical electrons not just identical, they completely loose their individuality. If, however, particles are indistinguishable then interchanging them should not produce any observable difference. This means that probability distribution corresponding to any two-particle state must remain the same when we exchange the particles. Mathematically speaking we can say that 1 2 2 1 1 2 2 1 ( , ) ( , ) P P σ σ σ σ = r r r r This means that the wave functions with interchanged particle’s coordinates may differ only by a phase factor ( 29 ( 29 1 2 2 1 1 2 2 1 , , i e δ σ σ σ σ ψ ψ = r r r r Interchanging the particle again, we will have ( 29 ( 29 1 2 1 2 2 1 2 1 2 , , i e δ σ σ σ σ ψ ψ = r r r r meaning that 0, δ π = and ( 29 ( 29 1 2 1 2 1 2 1 2 , , σ σ σ σ ψ ψ = ± r r r r
Bosons and fermions

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